Phase measurement profilometry, or PMP, is undergoing rapid development at present. Combined with defocused projection and advanced multiband phase shift (PS) algorithms, such as heterodyne and number theory, PMP with over thousand framerates has shown great potential in 3-D dynamic scene monitoring, e.g., manufacturing process monitoring and industrial digital twins. PMP’s outstanding advantages include a combination of high lateral resolution, axial precision, and speed; however, the widely used dual-band PS algorithms have limited axial precision compared to low-efficiency tri-band algorithms. In this article, we propose “3–2–1” PMP, which optimizes the existing dual-band approach by adding an extra fringe pattern for projection acquisition to achieve similar precision performance as tri-band PMP. Associated stabilization algorithms in dealing with fringe blur, nonlinear camera responses, phase-jump errors, and related impact factors have also been investigated. Comparative simulation and experiments show that the “3–2–1” approach produces higher precision performance, with a 42%–85% root-mean-square error lower than general dual-band PS methods and famous Fourier transform profilometry ( $\mu $ FTP), but using an equivalent projection pattern number of 6 or less.

“3-2-1” PMP: Adding an Extra Pattern to Dual-Band Phase-Shift Profilometry for Higher Precision 3D Imaging

Optics, renowned for its non-contact measurement capabilities, versatility, and high sensitivity, is recognized as a powerful tool driving the technological evolution of precision metrology [...] 

Precision Optical Metrology and Smart Sensing

This chapter focuses upon the use of both splines and machine-learning in prediction and the methodology for constructing splines in a predictive context. In the realm of predictive modeling, machine learning and splines represent two pivotal approaches that address the complexity of capturing nonlinear relationships within data. Machine learning excels in identifying intricate patterns and relationships through algorithms that learn from data, making it a powerful tool for prediction across vast datasets. However, its often opaque nature can pose challenges for interpretability. In contrast, splines offer a bridge between the simplicity of linear regression and the complexity of machine learning. By introducing cutpoints in the data, splines allow for flexible modeling of nonlinear trends, providing a clearer interpretation of how independent variables influence the dependent variable across different segments. This makes splines particularly valuable in multivariable regression contexts, where understanding the nuanced effects of covariates is crucial. While machine learning may deliver superior predictive power in some cases, splines provide a compelling balance of predictability and interpretability, especially in scenarios where understanding the underlying model is as important as the accuracy of predictions. 

Perspective Chapter: Enhancing Regression Analysis with Splines and Machine Learning – Evaluation of How to Capture Complex Non-Linear Multidimensional Variables

Parametric splines are popular tools for precision optical metrology of complex freeform surfaces. However, as a promising topologically unconstrained solution, existing T-spline fitting techniques, such as improved global fitting, local fitting, and split-connect algorithms, still suffer the problems of low computational efficiency, especially in the case of large data scales and high accuracy requirements. This paper proposes a speed-improved algorithm for fast, large-scale freeform point cloud fitting by stitching locally fitted T-splines through three steps of localized operations. Experiments show that the proposed algorithm produces a three-to-eightfold efficiency improvement from the global and local fitting algorithms, and a two-to-fourfold improvement from the latest split-connect algorithm, in high-accuracy and large-scale fitting scenarios. A classical Lena image study showed that the algorithm is at least twice as fast as the split-connect algorithm using fewer than 80% control points of the latter.

Stitching Locally Fitted T-Splines for Fast Fitting of Large-Scale Freeform Point Clouds

Stitching Locally Fitted T-Splines for Fast Fitting of Large-Scale Freeform Point-Clouds

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Stitching Locally Fitted T-Splines for Fast Fitting of Large-Scale Freeform Point Clouds

Stitching Locally Fitted T-Splines for Fast Fitting of Large-Scale Freeform Point-Clouds